Problem: $f'(x)=4f(x)$, and $f(0)=10$. Solve the equation. Choose 1 answer: Choose 1 answer: (Choice A) A $f(x)=4e^{10x}$ (Choice B) B $f(x)=e^{4x}$ (Choice C) C $f(x)=10e^{4x}$ (Choice D) D $f(x)=e^{10x}$
Solution: The general solution of equations of the form $f'(x)=kf(x)$ is $f(x)=C\cdot e^{kx}$ for some constant $C$. This can be found using separation of variables. In our case, $k=4$, so $f(x)=C\cdot e^{4x}$. Let's use the fact that $f(0)=10$ to find $C$ : $\begin{aligned} f(x)&=C\cdot e^{4x} \\\\ f(0)&=C\cdot e^{4\cdot 0} \gray{\text{Plug }x=0} \\\\ 10&=C\cdot e^{4\cdot 0} \gray{f(0)=10} \\\\ 10&=C \end{aligned}$ In conclusion, $f(x)=10e^{4x}$.